Simplify the following expression: $ x = \dfrac{a - 6}{-2a + 9} + \dfrac{9}{7} $
Solution: In order to add expressions, they must have a common denominator. Multiply the first expression by $\dfrac{7}{7}$ $ \dfrac{a - 6}{-2a + 9} \times \dfrac{7}{7} = \dfrac{7a - 42}{-14a + 63} $ Multiply the second expression by $\dfrac{-2a + 9}{-2a + 9}$ $ \dfrac{9}{7} \times \dfrac{-2a + 9}{-2a + 9} = \dfrac{-18a + 81}{-14a + 63} $ Therefore $ x = \dfrac{7a - 42}{-14a + 63} + \dfrac{-18a + 81}{-14a + 63} $ Now the expressions have the same denominator we can simply add the numerators: $x = \dfrac{7a - 42 - 18a + 81}{-14a + 63} $ $x = \dfrac{-11a + 39}{-14a + 63}$ Simplify the expression by dividing the numerator and denominator by -1: $x = \dfrac{11a - 39}{14a - 63}$